Virtual oscillator control of power electronics inverters

ABSTRACT

A system includes power electronics inverters connected in a network. The power electronics inverters can utilize measurements at local terminals, without a need to exchange information between other power electronics inverters.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application claims the benefit of U.S. Provisional Application Ser.No. 61/875,518, filed Sep. 9, 2013, which is incorporated in itsentirety herein.

BACKGROUND

A power inverter, or inverter, is an electronics device or circuitrythat changes direct current (DC) to alternating current (AC). Oneapplication of power electronics inverters are in microgrids. Microgridsare small-scale versions of the centralized electricity system. They canachieve specific local goals, such as reliability, carbon emissionreduction, diversification of energy sources, and cost reduction,established by the community being served. Like a bulk power grid, smartmicrogrids can generate, distribute, and regulate the flow ofelectricity to consumers. Smart microgrids are a way to integraterenewable resources on the community level and allow for customerparticipation in the electricity enterprise.

BRIEF DESCRIPTION OF THE DRAWINGS

In association with the following detailed description, reference ismade to the accompanying drawings, where like numerals in differentfigures can refer to the same element.

FIG. 1 is a circuit diagram of an exemplary oscillator structure.

FIG. 2 is a circuit diagram of an exemplary N oscillators interconnectedthrough an LTI network described by the admittance matrix Y(s).

FIG. 3 is a block diagram of an exemplary coupled oscillator system.

FIG. 4 is a block diagram of an exemplary equivalent differential systemwhich illustrates signal differences.

FIG. 5 is a circuit diagram of an exemplary single nonlinear oscillator.

FIGS. 6A and 6B are exemplary function diagrams for a deadzoneoscillator.

FIGS. 7A and 7B are exemplary phase-plots of steady-state limit-cyclesin the (a) deadzone and (b) Van der Pol oscillators for varying ε.

FIG. 8 is a circuit diagram of an exemplary N oscillator with no load.

FIG. 9 is a circuit diagram of an exemplary N oscillator with a linearload.

FIG. 10 is a plot of exemplary inverter output currents, voltages, andvoltage synchronization error when synchronization is guaranteed.

FIG. 11 is a plot of an exemplary evolution of state-variables duringstartup in the presence of a load.

FIG. 12 is a plot of exemplary inverter output currents, voltages, andvoltage synchronization error when synchronization is not guaranteed andnot achieved.

FIG. 13A is a circuit diagram of exemplary inverters that are controlledto emulate the exemplary oscillators in FIG. 13B.

FIG. 14 is a circuit diagram of an exemplary single-phase voltage sourceinverter with virtual oscillator control.

FIG. 15 is a circuit diagram of an averaged model of N connectedinverters and associated virtual oscillator controllers coupled througha microgrid network.

FIG. 16 is a block diagram representation of an exemplary coupledoscillator system when scaling gains are used.

FIG. 17 is a block diagram representation of an exemplary differentialsystem which illustrates signal differences and when scaling gains areused.

FIG. 18 is a circuit model used in control parameter selection.

FIG. 19 is a circuit diagram of an exemplary inverter controller with apre-synchronization circuit.

FIG. 20 is a circuit diagram of an exemplary inverter system.

FIG. 21 is an oscilloscope screenshot of exemplary measured inverteroutput currents and load voltage during system startup with a resistiveload.

FIGS. 22A and 22B are oscilloscope screenshots of resistive loadtransients showing exemplary inverter output currents and load voltageduring (a) load step-up and (b) load step-down.

FIGS. 23A and 23B are oscilloscope screenshots of inverter outputcurrents and load voltage when an inverter is (a) removed and (b) addedin the presence of a resistive load.

FIGS. 24A and 24B are circuit diagrams of exemplary circuits for (a)linear RLC load and (b) nonlinear diode bridge rectifier load.

FIG. 25 is an oscilloscope screenshot of exemplary measured inverteroutput currents and load voltage during system startup with an RLC load.

FIGS. 26A and 26B are oscilloscope screenshots of RLC load transientshowing inverter output currents and load voltage when RL load branch is(a) added and (b) removed.

FIG. 27 is an oscilloscope screenshot of exemplary measured inverteroutput currents and load voltage in the presence of a diode bridgerectifier load.

FIGS. 28A and 28B are oscilloscope screenshots of inverter outputcurrents and load voltage when connected to a diode bridge rectifierload and an inverter is (a) removed and (b) added.

FIG. 29 is an oscilloscope screenshot of exemplary measured inverteroutput currents and load voltage in the presence of two fan loads.

FIGS. 30A and 30B are oscilloscope screenshots of mechanical loadtransients showing inverter output currents and load voltage during load(a) step-up and (b) load step-down.

FIG. 31 is a circuit diagram of an exemplary single-phase, constantpower load.

FIG. 32 is an oscilloscope screenshot of exemplary simulated invertercurrents, load voltage, and power consumed by the dc load.

FIG. 33 is a circuit diagram of an exemplary system of inverters withvirtual oscillator control in (a) and exemplary system of coupledoscillators in (b).

FIGS. 34 and 35 are phase diagrams of an exemplary coordinatetransformation of three-phase signals.

FIG. 36 is a circuit diagram of an exemplary deadzone oscillator forthree-phase signals.

FIG. 37 is a circuit diagram of an exemplary inverter controller forthree-phase signals.

DETAILED DESCRIPTION

A system, method and/or device, etc. are described to control acollection of power electronics inverters. The power electronicsinverters only require their localized measurements. No localmeasurement information, such as system frequency or network-levelset-points, needs to be exchanged across the system of inverters. Theindividual inverters are able to utilize local measurements anddetermine what to do such that when a large collection of inverters areconnected together they work together cohesively as a system. Therefore,power supplies can be synchronized without the need for communication,e.g., an exchange of information.

Synchronization is described for nonlinear oscillators coupled through alinear time-invariant theory (LTI) network. Synchronization ofdistributed oscillator systems can be utilized in several areas,including neural processes, coherency in plasma physics, communications,and electric power systems. A condition is described herein for theglobal asymptotic synchronization of a class of identical nonlinearoscillators coupled through an electrical network with LTI elements,e.g., resistors, capacitors, inductors, and transformers. Forexplanation purposes, in one implementation, symmetric networksincluding oscillators connected to a common node through identicalbranch impedances are described. For this type of network, thesynchronization condition is independent of: i) the load impedance, andii) the number of oscillators in the network. The results can be used toformulate a control paradigm for the coordination of inverters serving apassive electrical load in a microgrid. For purposes of explanation theinverters are connected in parallel, but non-parallel connections canalso be implemented with the systems and methods.

Because power networks generally include LTI circuit elements(resistors, capacitors, inductors, and transformers), passivity-basedsynchronization analysis can be difficult to apply in this setting.Additionally or alternatively, the notion of L₂ input-output stabilitycan be used to analyze synchronization in systems coupled through an LTInetwork. Given the differential equations of the original system, atransformation is performed such that an equivalent system based onsignal differences is formulated. Referred to herein as a differentialsystem. If the resulting differential system is input-output stable,signal differences decay to zero and oscillator synchronization results.One advantage of this technique is that the analyst does not need toformulate a storage function. Using established L₂ input-outputstability methods, a sufficient condition for synchronization can beattained. The results described herein can derive from L₂ methodsbecause they facilitate analysis of LTI power networks.

For explanation purposes, the results are focused on coupled oscillatorsystems. A sufficient global asymptotic synchronization condition isderived for a class of identical nonlinear oscillators coupled throughan LTI network. For the particular network topology where theoscillators are connected to a common node through identical branchimpedances, the synchronization condition is independent of the numberof oscillators and the load impedance. From an application perspective,the results can be applied to the coordination of inverters in asingle-phase microgrid to achieve a control and design paradigm that isrobust (e.g., independent of load) and modular (e.g., independent ofnumber of inverters). Results are presented with a level of generalitysuch that they can be applied to a variety of implementations. Resultsderived can be applied towards the design and implementation of amicrogrid with N parallel inverters, described below.

For the N-tuple (u₁, . . . , u_(N)), denote u=[u₁, . . . , u_(N)]^(T) tobe the corresponding column vector, where ^(T) indicates transposition.The N-dimensional column vectors of all ones and all zeroes are denotedby 1 and 0, respectively. N×N matrices can be denoted by U and they canbe diagonalized as U=Q

Q⁻¹, where Λ denotes the diagonal matrix of eigenvalues and the columnvectors of Q are the corresponding eigenvectors. The Laplace transformof the continuous-time function f(t) is denoted by f(s), where s=ρ+jω isa complex number, and j=√{square root over (−1)}. Transfer functions aredenoted by lower-case z(s), and transfer matrices are denoted byupper-case Z(s). Unless stated otherwise, Z(s)=z(s) I_(N), where I_(N)is the N×N identity matrix. The Euclidean norm of a real or complexvector, u, is denoted by ∥u∥₂ and is determined as:∥u∥ ₂ =√{square root over (u*u)}  (1)

where * indicates the conjugate transpose. If u is real, then u*=u^(T).For some continuous-time function u(t), u:[0, ∞)→

^(N), the

₂ norm of u is determined as∥u

=√{square root over (∫₀ ^(∞) u(t)^(T) u(t)dt)},  (2)

and the space of piecewise-continuous and square-integrable functionswhere ∥u∥_(L) ₂ <∞ is denoted by

₂. If uε

₂ then u is said to be bounded. A causal system, H, with input u andoutput y, is said to be finite-gain

₂ stable if there exist finite, non-negative constants, γ and η, suchthat∥y∥ _(L) ₂ =∥H(u)∥_(L) ₂ ≦γ∥u∥ _(L) ₂ +η, ∀uεL ₂.  (3)

The smallest value of γ for which there exists a η such that (3) issatisfied is called the L₂ gain of the system. The L₂ gain of H, denotedas γ (H), provides a measure of the largest amplification applied to theinput signal, u, as it propagates through the system H. Intuitively, (3)can be understood as stating that norm of the output, H(u), cannot belarger than the linearly scaled norm of the input u. Hence, the systemis described as being input-output stable when the L₂ gain of H isfinite.

If H is linear and can be represented by the matrix of transferfunctions H(s) such that H(s)ε

^(N×N), the L₂ gain of H is equal to its H-infinity norm, denoted by ∥H∥∞, and determined as

$\begin{matrix}{{{\gamma(H)} = {{H}_{\infty} = {\sup\limits_{\omega \in \bullet}\frac{{{{H({j\omega})}{u({j\omega})}}}_{2}}{{{u({j\omega})}}_{2}}}}},} & (4)\end{matrix}$

where ∥u(jω)∥₂=1, provided that all poles of H(s) have strictly negativereal parts. Because (4) is the ratio of the output to input norms, γ (H)gives a measure of the largest amplification of the vector u when it ismultiplied by the matrix H(s). Stated alternatively, γ (H) is thelargest singular value of the matrix H(s). Note that if H(s) is asingle-input single-output transfer function such that H(s)ε

, then γ (H)=

${H}_{\infty} = {\sup\limits_{\omega \in \bullet}{{{H({j\omega})}}_{2}.}}$

A classical result that can be useful in showing synchronization isBarbalat's lemma. Consider the continuous function φ:[0, ∞)→

. Barbalat's lemma states that if lim_(t→∞)∫₀ ^(t)φ(τ)dτ<∞, then

$\begin{matrix}{{\lim\limits_{t\rightarrow\infty}{\phi(t)}} = 0.} & (5)\end{matrix}$

This property can be used to show that all signal differences decay tozero when the system meets the sufficient condition for synchronization.The electrical networks under study have underlying graphs that areundirected and connected. The corresponding Laplacian matrix, denoted byΓε

^(N×N), has the following properties:

1. rank(Γ)=N−1

2. The eigenvalues of Γ (ordered in ascending order by magnitude) aredenoted by λ₁<λ₂< . . . <λ_(N), where λ₁=0.

3. Γ is symmetric with row and column sums equal to zero such thatΓ1=Γ^(T)1=0.

4. The eigenvector q₁ (corresponding to λ₁=0) is given by

$q_{1} = {\frac{1}{\sqrt{N}}1.}$

5. The Laplacian can be diagonalized as Γ=Q

Q^(T), where it follows that Q⁻¹=Q^(T) because Γ=Γ^(T).

A useful construct that can be employed to compare individual oscillatoroutputs with the average of all N oscillator outputs is the projectormatrix, Π, determined as

$\begin{matrix}{\Pi = {I_{N} - {\frac{1}{N}11^{T}}}} & (6)\end{matrix}$

For some vector uε

^(N), denote ũ=Πu, and refer to ũ; as the corresponding differentialvector. A causal system, H, with input u and output y, is said to bedifferentially finite-gain L₂ stable if there exist finite, non-negativeconstants, {tilde over (γ)} and {tilde over (η)} such that∥{tilde over (γ)}∥_(L) ₂ ≦{tilde over (γ)}∥ũ∥ _(L) ₂ +{tilde over (η)},∀ũεL ₂.  (7)

Where {tilde over (y)}=Πy. The smallest value of for {tilde over (y)}for which there exists a {tilde over (η)}; such that (7) is satisfied,is called the differential L₂ gain of the system and is denoted as{tilde over (γ)}(H). The differential L₂ gain provides a measure of theamplification of signal differences as they propagate through a system.

Conditions for global asymptotic synchronization are described. Asystem-level description of the coupled nonlinear oscillators isprovided. The projector matrix is used to derive a corresponding systembased on signal differences. Equipped with the differential systemdescription, a sufficient condition can be presented for globalasymptotic synchronization of the coupled oscillators.

FIG. 1 is a circuit diagram of an exemplary oscillator structure. Theoscillator has: i) a linear subsystem comprised of passive circuitelements with impedance z_(osc)(s), and ii) a nonlinearvoltage-dependent current source, g(v). The source g(v) is required tobe continuous and differentiable, and additionally requires:

$\begin{matrix}{\sigma:={{\sup\limits_{v \in \bullet}{{\frac{\mathbb{d}}{\mathbb{d}v}{g(v)}}}} < \infty}} & (8)\end{matrix}$

In other words, the slope of g(v) with respect to the oscillator voltageis bounded.

Consider a system in which N such oscillators are coupled through apassive electrical LTI network, and the coupling is captured through:i(s)=Y(s)v(s)  (9)

where i(s)=[i₁(s), . . . , i_(N)(s)]^(T) is the vector of oscillatoroutput currents, v(s)=[v₁(s), . . . , v_(N)(s)]^(T) is the vector ofoscillator terminal voltages, and Y(s) is the network admittance matrixof the general form:Y(s)=α(s)I _(N)+β(s)Γ,  (10)

where α(s), β(s)ε

, and Γ is the network Laplacian with the properties described inSection 1. The admittance matrix of the microgrid network underconsideration does have the form shown in (10). Conceptually, theadmittance, α(s), in the first term of (10) can be understood as thelocal load observed from the output of each oscillator while the secondterm, β(s)Γ, accounts for the interaction between units. As the systemsynchronizes, the interaction between oscillators decays to zero and theeffective output impedance observed from the each oscillator is equal toα(s)⁻¹.

FIG. 2 is a circuit diagram of an exemplary N oscillators interconnectedthrough a coupling network described by the admittance matrix Y(s). Theterminal voltage of the j^(th) oscillator, v_(j)(s), can be expressedas:v _(j)(s)=z _(osc)(s)(i _(srcj)(s)−i _(j)(s)),∀j=1, . . . ,N.  (11)

Writing all terminal voltages in matrix form yields:

$\begin{matrix}\begin{matrix}{{v(s)} = {{Z_{osc}(s)}\left( {{i_{src}(s)} - {i(s)}} \right)}} \\{{= {{{Z_{osc}(s)}{i_{src}(s)}} - {{Z_{osc}(s)}{Y(s)}{v(s)}}}},}\end{matrix} & (12)\end{matrix}$

where Z_(osc)(s)=z_(osc) (s)·I_(N)ε

^(N×N), i_(src)(s)=[i_(src1)(s), . . . , i_(srcN)(s)]^(T), and in thesecond line of (12), i(s)=Y(s)v(s) from (9) has been substituted. v(s)can be isolated from (12) as follows:

$\begin{matrix}\begin{matrix}{{v(s)} = {\left( {I_{N} + {{Z_{osc}(s)}{Y(s)}}} \right)^{- 1}{Z_{osc}(s)}{i_{src}(s)}}} \\{{= {{F\left( {{Z_{osc}(s)},{Y(s)}} \right)}{i_{src}(s)}}},}\end{matrix} & (13)\end{matrix}$

where F:□^(N×N)×□^(N×N)→□^(N×N) is called the linear fractionaltransformation. In general, for some A,B of appropriate dimension anddomain, the linear fractional transformation is determined as:F(A,B):=(I _(N) +AB)⁻¹ A.  (14)

the system of coupled oscillators admits the compact block-diagramrepresentation in FIG. 3), where the linear and nonlinear portions ofthe system are compartmentalized by F(Z_(osc)(s),Y (s)) and g(v)=[g(v₁),. . . , g(v_(N))]^(T), respectively.

FIG. 3 is a block diagram of an exemplary coupled oscillator system.g(v) is a vector which captures the input-output relation of all thenonlinear circuit elements in the oscillators. Asymptoticsynchronization can be utilized in the network introduced above.Oscillator synchronization is determined as:

$\begin{matrix}{{{{\lim\limits_{t\rightarrow\infty}{v_{j}(t)}} - {v_{k}(t)}} = {0{\forall j}}},{k = 1},\ldots\mspace{14mu},{N.}} & (15)\end{matrix}$

For ease of analysis, it can be useful to transform to a coordinatesystem based on signal differences. Subsequently, such a system can bereferred to as the corresponding differential system. Towards this end,the projector matrix determined in (6) has the following property:

$\begin{matrix}{{{v(t)}^{T}{v(t)}} = {{\left( {\Pi\;{v(t)}} \right)^{T}\left( {\Pi\;{v(t)}} \right)} = {\frac{1}{2\; N}{\sum\limits_{j = 1}^{N}\;{\sum\limits_{k = 1}^{N}\;{\left( {{v_{j}(t)} - {v_{k}(t)}} \right)^{2}.}}}}}} & (16)\end{matrix}$

Therefore, the synchronization condition in (15) is equivalent torequiring v(t)=Πv(t)→0 as t→∞.

The corresponding differential system can be derived. The differentialterminal-voltage vector, {tilde over (v)}(s), can be expressed as:

$\begin{matrix}\begin{matrix}{{v(s)} = {{\Pi\;{v(s)}} = {\Pi\left( {{Z_{osc}(s)}\left( {{i_{src}(s)} - {i(s)}} \right)} \right)}}} \\{= {{Z_{osc}(s)}\left( {{\Pi\;{i_{src}(s)}} - {\Pi\;{Y(s)}{v(s)}}} \right)}} \\{{= {{Z_{osc}(s)}\left( {{i_{src}(s)} - {{Y(s)}{v(s)}}} \right)}},}\end{matrix} & (17)\end{matrix}$

where in the first line, v(s) from (12) has been substituted, and in thesecond line, the relation i(s)=Y(s)v(s) from (9) is used and the factthat ΠZ_(osc)(s)=Πz_(osc) (s)I_(N)=z_(osc)(s)I_(N)Π=Z_(osc)(s)Π. Thelast line follows from the fact that the projector and admittancematrices commute, e.g., ΠY(s)=Y(s)Π. To prove this, for the class ofadmittance matrices given by (10), note that:

$\begin{matrix}\begin{matrix}{{\Pi\;{Y(s)}} = {\Pi\left( {{{\alpha(s)}I_{N}} + {{\beta(s)}\Gamma}} \right)}} \\{= {{{\alpha(s)}I_{N}\Pi} + {{\beta(s)}{\Pi\Gamma}}}} \\{= {{{\alpha(s)}{IN}\;\Pi} + {{\beta(s)}\left( {I_{N} - {{1/N}\; 11^{T}}} \right)\Gamma}}} \\{= {{{\alpha(s)}I_{N}\Pi} + {{\beta(s)}\left( {{\Gamma\; I_{N}} - {{1/N}\;{\Gamma 11}^{T}}} \right)}}} \\{{= {{\left( {{{\alpha(s)}I_{N}} + {{\beta(s)}\Gamma}} \right)\Pi} = {{Y(s)}\Pi}}},}\end{matrix} & (18)\end{matrix}$

where the row and column sums of Γ are zero is used, which implies 11^(T) Γ=0 0^(T)=Γ1 1^(T). {tilde over (v)}(s) in (17) can now beisolated as follows:

$\begin{matrix}\begin{matrix}{{v(s)} = {\left( {I_{N} + {{Z_{osc}(s)}{Y(s)}}} \right)^{- 1}{Z_{osc}(s)}{i_{src}(s)}}} \\{= {{F\left( {{Z_{osc}(s)},{Y(s)}} \right)}{{i_{src}(s)}.}}}\end{matrix} & (19)\end{matrix}$

Notice the similarity between (19) and (13). Note that the linearfractional transformation also maps i_(src)(s) to {tilde over (v)}(s).

Determine a map {tilde over (g)}; that captures the impact of g(v) inthe corresponding differential system as follows:{tilde over (g)}:{tilde over (v)}→{tilde over (ι)} _(src)  (20)

A complete description of the equivalent differential system has beenattained. The system admits the block diagram representation in FIG. 4,where, as in FIG. 3, the linear and nonlinear subsystems arecompartmentalized using

(•,•) and {tilde over (g)}, respectively.

FIG. 4 is a block diagram of an exemplary equivalent differentialsystem. Conditions are derived that ensure global asymptoticsynchronization in the sense of (15) for the system of oscillatorsdescribed above. Before Theorem 1 is stated, a lemma is first presentedwhich gives an upper bound on the differential

₂ gain of the nonlinearity g(·).

Lemma 1. The differential

₂ gain of g is finite and upper bounded by a such that:

$\begin{matrix}{{{\gamma(g)} \leq \sigma} = {{\sup\limits_{v \in \bullet}{{\frac{\mathbb{d}}{\mathbb{d}v}{g(v)}}}} < {\infty.}}} & (21)\end{matrix}$

By definition of σ, for any pair of terminal voltages v_(j) and v_(k),and the corresponding source currents i_(srcj) and i_(srck), wherej,kε{1, . . . , N}, the mean-value theorem can be applied to give:

$\begin{matrix}\left. {\sigma \geq \frac{{{i_{srcj}(t)} - {i_{srck}(t)}}}{{{v_{j}(t)} - {v_{k}(t)}}}}\Rightarrow{{\sigma^{2}\left( {{v_{j}(t)} - {v_{k}(t)}} \right)}^{2} \geq {\left( {{i_{srcj}(t)} - {i_{srck}(t)}} \right)^{2}.}} \right. & (22)\end{matrix}$

Summing over all indices, j, kε{1, . . . , N} in (22) yields

$\begin{matrix}{{{\sigma^{2}{\sum\limits_{j = 1}^{N}\;{\sum\limits_{k = 1}^{N}\;\left( {{v_{j}(t)} - {v_{k}(t)}} \right)^{2}}}} \geq {\sum\limits_{j = 1}^{N}\;{\sum\limits_{k = 1}^{N}\;\left( {{i_{srcj}(t)} - {i_{srck}(t)}} \right)^{2}}}},} & (23)\end{matrix}$which can be rearranged and simplified as follows

$\begin{matrix}{\sigma \geq {\sqrt{\frac{\sum\limits_{j = 1}^{N}\;{\sum\limits_{k = 1}^{N}\;\left( {{i_{srcj}(t)} - {i_{srck}(t)}} \right)^{2}}}{\sum\limits_{j = 1}^{N}\;{\sum\limits_{k = 1}^{N}\;\left( {{v_{j}(t)} - {v_{k}(t)}} \right)^{2}}}}.}} & (24)\end{matrix}$Since (24) holds for any set of terminal voltages, this implies

$\begin{matrix}{{\sigma \geq {\sup\limits_{v \in \bullet^{N}}\sqrt{\frac{\frac{1}{2\; N}{\sum\limits_{j = 1}^{N}\;{\sum\limits_{k = 1}^{N}\;\left( {{i_{srcj}(t)} - {i_{srck}(t)}} \right)^{2}}}}{\frac{1}{2\; N}{\sum\limits_{j = 1}^{N}\;{\sum\limits_{k = 1}^{N}\;\left( {{v_{j}(t)} - {v_{k}(t)}} \right)^{2}}}}}}},} & (25)\end{matrix}$which can be rewritten compactly using the projector-matrix notation in(16) as

$\begin{matrix}{\sigma \geq {\sup\limits_{v \in \bullet^{N}}{\sqrt{\frac{{i_{src}(t)}^{T}{i_{src}(t)}}{{v(t)}^{T}{v(t)}}}.}}} & (26)\end{matrix}$

By definition of the differential

₂ gain,

$\begin{matrix}\begin{matrix}{{\overset{\sim}{\gamma}(g)} = {\sup\limits_{v \in \bullet^{N}}\frac{{i_{src}}_{L_{2}}}{{v}_{L_{2}}}}} \\{= {\sup\limits_{v \in \bullet^{N}}{\frac{\sqrt{\int_{0}^{\infty}{{{\overset{\bullet}{i}}_{src}(t)}^{T}{{\overset{\bullet}{i}}_{src}(t)}\ {\mathbb{d}t}}}}{\sqrt{\int_{0}^{\infty}{{\overset{\bullet}{v}(t)}^{T}{\overset{\bullet}{v}(t)}\ {\mathbb{d}t}}}}.}}}\end{matrix} & (27)\end{matrix}$

Applying (26) in the definition above,

$\begin{matrix}{{{{{\bullet\gamma}(g)} \leq {\sup\limits_{v \in \bullet^{N}}\frac{\sqrt{\sigma^{2}{\int_{0}^{\infty}{{\overset{\bullet}{v}(t)}^{T}{\overset{\bullet}{v}(t)}\ {\mathbb{d}t}}}}}{\sqrt{\int_{0}^{\infty}{{\overset{\bullet}{v}(t)}^{T}{\overset{\bullet}{v}(t)}\ {\mathbb{d}t}}}}}} = {\sigma < \infty}},} & (28)\end{matrix}$which completes the proof.

A sufficient condition for global asymptotic synchronization can beimplemented in the network of oscillators described in Section 2.1.

Theorem 1: The network of N oscillators coupled through (9) with theadmittance matrix in (10), synchronizes in the sense of (15), if∥F(ζ(s),β(s)λ₂∥_(∞)σ<1,  (29)where λ₂ is the smallest positive eigenvalue of Γ, and

$\begin{matrix}{{\zeta(s)}:={\frac{z_{osc}(s)}{1 + {{\alpha(s)}{z_{osc}(s)}}}.}} & (30)\end{matrix}$

Consider the block-diagram of the differential system in FIG. 4. Denotethe differential

₂ gain of the linear fractional transformation by {tilde over(γ)}(F(Z_(osc)(s),Y(s))). The finite-gain differential

₂ stability of F(Z_(osc)(s),Y (s)) gives∥{tilde over (v)}∥ _(L) ₂ ≦{tilde over (γ)}(F(Z _(osc)(s),Y(s)))∥ĩ_(src)∥_(L) ₂ +{tilde over (η)},  (31)for some non-negative {tilde over (η)}. Applying (21) from Lemma 1, itfollows that∥ĩ _(src)∥_(L) ₂ ≦σ∥{tilde over (v)}∥ _(L) ₂ .  (32)Combining (31) and (32) yields∥{tilde over (v)}∥ _(L) ₂ ≦{tilde over (γ)}(F(Z _(osc)(s),Y(s)))σ∥{tildeover (v)}∥ _(L) ₂ +{tilde over (η)}.  (33)

It can be required that{tilde over (γ)}(F(Z _(osc)(s),Y(s)))·σ<1  (34)

Isolating ∥{tilde over (v)}

leads to

$\begin{matrix}{{{\overset{\sim}{v}}_{\mathcal{L}\; 2} \leq \frac{\eta}{1 - {{\gamma\left( {F\left( {{Z_{osc}(s)},{Y(s)}} \right)} \right)}\sigma}}},} & (35)\end{matrix}$

which implies that {tilde over (v)}; ε

₂. It follows from Barbalat's lemma that

$\begin{matrix}{{{\lim\limits_{t\rightarrow\infty}\mspace{14mu}{\bullet\;{v(t)}}} = {\left. 0\Rightarrow{{\lim\limits_{t\rightarrow\infty}\mspace{14mu}{v_{j}(t)}} - {v_{k}(t)}} \right. = {0\mspace{14mu}{\forall j}}}},{k = 1},\ldots\mspace{14mu},{N.}} & (36)\end{matrix}$

That is, if the system of oscillators satisfies the condition in (34),global asymptotic synchronization can be guaranteed.

The result in (29) can now be derived by showing {tilde over(γ)}(F(Z_(osc)(s),Y(s))) equals ∥F(ζ(s),β(s)λ₂)∥. From the definition ofthe linear fractional transformation in (14), and the general form ofthe admittance matrix in (10), note that

$\begin{matrix}\begin{matrix}{{F\left( {{Z_{osc}(s)},{Y(s)}} \right)} = {\left( {I_{N} + {{Z_{osc}(s)}{Y(s)}}} \right)^{- 1}{Z_{osc}(s)}}} \\{= {\left( {I_{N} + {{Z_{osc}(s)}\left( {{{\alpha(s)}I_{N}} + {{\beta(s)}\Gamma}} \right)}} \right)^{- 1}{Z_{osc}(s)}}} \\{= {\left( {{\left( {1 + {{\alpha(s)}{z_{osc}(s)}}} \right)I_{N}} + {{z_{osc}(s)}{\beta(s)}\Gamma}} \right)^{- 1}{Z_{osc}(s)}}} \\{= {\left( {I_{N} + {\frac{z_{osc}(s)}{1 + {{\alpha(s)}{z_{osc}(s)}}}{\beta(s)}\Gamma}} \right)^{- 1}\frac{{z_{osc}(s)}I_{N}}{1 + {{\alpha(s)}{z_{osc}(s)}}}}} \\{= {{F\left( {{{\zeta(s)}I_{N}},{{\beta(s)}\Gamma}} \right)}.}}\end{matrix} & (37)\end{matrix}$

Because F(ζ(s)I_(N),β(s)Γ) is a linear system, it follows that thedifferential

₂ gain of F(Z_(osc)(s),Y (s)) can be calculated using the

-infinity norm. By definition of the

-infinity norm and differential

₂ gain, it follows that

$\begin{matrix}{{{{F\left( {{{\zeta(s)}I_{N}},{{\beta(s)}\Gamma}} \right)}} = {{\sup\limits_{\omega \in \bullet}\mspace{14mu}\frac{{{\overset{\sim}{v}\left( {j\;\omega} \right)}}_{2}}{{{{\overset{\sim}{i}}_{src}\left( {j\;\omega} \right)}}_{2}}} = {{\sup\limits_{\omega \in \bullet}\mspace{14mu}\frac{{{\left( {I_{N} + {{\zeta\left( {j\;\omega} \right)}{\beta\left( {j\;\omega} \right)}\Gamma}} \right)^{- 1}{\zeta\left( {j\;\omega} \right)}{{\overset{\sim}{i}}_{src}\left( {j\;\omega} \right)}}}_{2}}{{{{\overset{\sim}{i}}_{src}\left( {j\;\omega} \right)}}_{2}}} = {\sup\limits_{\omega \in \bullet}\mspace{14mu}\frac{{{{Q\left( {I_{N} + {{\zeta\left( {j\;\omega} \right)}{\beta\left( {j\;\omega} \right)}\Lambda}} \right)}^{- 1}{\zeta\left( {j\;\omega} \right)}Q^{T}{{\overset{\sim}{i}}_{src}\left( {j\;\omega} \right)}}}_{2}}{{{Q^{T}{{\overset{\sim}{i}}_{src}\left( {j\;\omega} \right)}}}_{2}}}}}},} & (38)\end{matrix}$

where Γ was diagonalized as Γ=Q

Q^(T) in the second line above. Two observations can be made to simplify(38):

i) The first column of Q is given by

$q_{1} = {\frac{1}{\sqrt{N}}1.}$Furthermore,

${1^{T}\prod} = {{1^{T}\left( {I_{N} - {\frac{1}{N}11^{T}}} \right)} = {{1^{T} - {\frac{1}{N}\left( {1^{T}1} \right)1^{T}}} = {0^{T}.}}}$

Therefore, the vector Q^(T){tilde over (ι)}_(src)(s)=Q^(T)Πi_(src) (s)is given byQ ^(T) ĩ _(src)(s)=Q ^(T) Πi _(src)(s)=[0,D(s)]^(T),  (39)

where D(s) ε

^(N-1×1) is made up of the non-zero elements of the vectorQ^(T)Πi_(src)(s).

ii) Denote the diagonal matrix with diagonal entries comprised of thenon-zero eigenvalues of Γ by Λ_(N-1), i.e., Λ_(N-1)=diag {λ₂, . . . ,λ_(N)}ε□^(N-1×N-1),

Using the two observations highlighted above, (38) can now be simplifiedas

$\begin{matrix}{{{{F\left( {{{\zeta(s)}I_{N}},{{\beta(s)}\Gamma}} \right)}} = {{\sup\limits_{\omega \in \bullet}\mspace{14mu}\frac{{{\left( {I_{N - 1} + {{\zeta\left( {j\;\omega} \right)}{\beta\left( {j\;\omega} \right)}\Lambda_{N - 1}}} \right)^{- 1}{\zeta\left( {j\;\omega} \right)}{D\left( {j\;\omega} \right)}}}_{2}}{{{D\left( {j\;\omega} \right)}}_{2}}} = {{\sup\limits_{{j = 2},\ldots\mspace{14mu},N}\mspace{14mu}\left( {\sup\limits_{\omega \in \bullet}\left( \frac{{D^{*}\left( {j\;\omega} \right)}\left( {1 + {{\zeta\left( {j\;\omega} \right)}{\beta\left( {j\;\omega} \right)}\lambda_{j}}} \right)^{- 2}{\zeta^{2}\left( {j\;\omega} \right)}{D\left( {j\;\omega} \right)}}{{D^{*}\left( {j\;\omega} \right)}{D\left( {j\;\omega} \right)}} \right)}^{\frac{1}{2}} \right)} = {{\sup\limits_{{j = 2},\ldots\mspace{14mu},N}\mspace{14mu}\left( {{\sup\limits_{\omega \in \bullet}\left( {1 + {{\zeta\left( {j\;\omega} \right)}{\beta\left( {j\;\omega} \right)}\lambda_{j}}} \right)}^{- 1}{\zeta\left( {j\;\omega} \right)}} \right)} = {{\sup\limits_{{j = 2},\ldots\mspace{14mu},N}\mspace{14mu}{{F\left( {{\zeta(s)},{{\beta(s)}\lambda_{j}}} \right)}}_{\infty}} = {{F\left( {{\zeta(s)},{{\beta(s)}\lambda_{2}}} \right)}}_{\infty}}}}}},} & (40)\end{matrix}$

where the last equality follows from the fact that ∥F(ζ(s),β(s)λ)∥ is adecreasing function of λ. From (34) and (40), (29) is a sufficientcondition for global asymptotic synchronization.

The proof for Theorem 1 can be thought of as being based on theclosed-loop block-diagram of the differential system in FIG. 17,described below. The

₂ gain provides a measure of the largest amplification imparted by asystem as a signal propagates through it. Thus, if the product of thedifferential

₂ gains, {tilde over (γ)}(F(Z_(osc)(s),Y(s))) and {tilde over (γ)}(g),is less than 1, then the differential vectors, {tilde over (v)}; and{tilde over (ι)}_(src), both decay to zero and oscillatorsynchronization results. It can equivalently be stated that because thedifferential system in FIG. 17 is stable, the differential vectors tendtowards zero.

The oscillator model which can form the basis of the inverter controlcan be described and the coupling network of a microgrid with N parallelinverters can be characterized. When Theorem 1 is applied to the systemof interest, the synchronization criterion is independent of the numberof oscillators and load parameters.

Before describing the oscillator, Liénard's theorem is stated below. Thetheorem can be used to establish the existence of a stable and uniquelimit cycle in the particular oscillator under study.

Theorem 2. Consider the system{umlaut over (v)}+r(v){dot over (v)}+m(v)=0,  (41)

where v: [0, ∞)→□ and r(v),m(v): R→R are differentiable with respect tov.

The functions, r(v) and m(v), are even and odd, respectively. Inaddition, determineR(v):=∫₀ ^(v) r(τ)dτ.  (42)

The system in (41) has a unique and stable limit cycle if: i) m(v)>0∀v>0, ii) R(v) has one positive zero for some v=p, iii) R(v)<0 when0<v<p, and iv) R(v) monotonically increases for

${\lim\limits_{v\rightarrow\infty}\mspace{14mu}{R(v)}} = {\infty.}$

FIG. 5 is a circuit diagram of an exemplary single nonlinear oscillator.The deadzone oscillator shown in FIG. 5 is used, in which the linearsubsystem is includes a RLC circuit with impedance:z _(osc)(s)=R∥sL∥(sC)⁻¹  (43)

and the nonlinear current source is given by:g(v)=ƒ(v)−σv,  (44)

where f(•) is a continuous, differentiable deadzone function with slope2 σ, and f(v)≡0 for vε(−φ,+φ), as illustrated in FIG. 6(a). The functiong(v), which is plotted in FIG. 6(b), resembles a piecewise linearfunction. As illustrated in FIG. 6(b), the nonlinear current source,g(v), acts as a power source for v<2 φ and as a dissipative element whenv>2 φ. The functions (a) f(v) and (b) g(v) illustrated for the deadzoneoscillator. For the deadzone oscillator,

${\sup\limits_{v \in \bullet}\mspace{14mu}{{\frac{\mathbb{d}\;}{\mathbb{d}v}{g(v)}}}} = {\sigma.}$A Van der Pol oscillator utilizes a cubic nonlinearity instead of adeadzone nonlinearity as described here.

Using Kirchhoff's voltage and current laws, the terminal voltage of thedeadzone oscillator can be determined as:

$\begin{matrix}{{{{LC}\frac{\mathbb{d}^{2}v}{\mathbb{d}t^{2}}} + {{L\left( {\frac{\mathbb{d}{f(v)}}{\mathbb{d}v} + \frac{1}{R} - \sigma} \right)}\frac{\mathbb{d}v}{\mathbb{d}t}} + v} = 0.} & (45)\end{matrix}$

Number (45) can be rewritten by expressing the derivatives of v withrespect to τ=t/√{square root over (LC)} to get:

$\begin{matrix}{{{\overset{¨}{v} + {\sqrt{\frac{L}{C}}\left( {\frac{\mathbb{d}{f(v)}}{\mathbb{d}v} + \frac{1}{R} - \sigma} \right)\overset{.}{v}} + v} = 0},} & (46)\end{matrix}$

which is of the form in (41), with:

$\begin{matrix}\left\{ {\begin{matrix}{{m(v)} = v} \\{{r(v)} = {\sqrt{\frac{L}{C}}\left( {\frac{\mathbb{d}{f(v)}}{\mathbb{d}v} + \frac{1}{R} - \sigma} \right)}}\end{matrix}.} \right. & (47)\end{matrix}$

For the case σ>1/R, m(v), r(v), and R(v) satisfy the conditions inLinard's theorem, implying that the deadzone oscillator has a stable andunique limit cycle. The steady-state limit cycles of the deadzoneoscillator are plotted for different values of

$\overset{\backprime}{o} = {\sqrt{\frac{L}{C}}\left( {\sigma - \frac{1}{R}} \right)}$in FIG. 7(a). For comparison, the limit cycles of the Van der Poloscillator for the same set of parameters is shown in FIG. 7(b). Whenε<<1, the steady-state oscillation can have a frequency approximatelyequal to

$\frac{1}{\sqrt{LC}}.$For small values of ε, the phase-plot resembles a unit circle, and as aresult, the voltage oscillation approximates an ideal sinusoid in thetime-domain.

The oscillation results from a periodic energy exchange between thepassive RLC circuit and nonlinear element, g(v), at the RLC resonantfrequency,

$\omega_{o} = {\frac{1}{\sqrt{LC}}.}$The piecewise nonlinearity in FIG. 6(b) acts as a dissipative circuitelement when the i-v curve lies in quadrants I and III and as a powersource when in quadrants II and IV. As a result, the nonlinear currentsource injects power into the system for small values of v, anddissipates power for large values of v. The overall tendency is forsmall oscillations to grow while large oscillations are damped such thata unique steady-state oscillation of some intermediate amplitude isreached.

In the following example, the objective can be to design a 60 Hzdeadzone oscillator. The parameters R, L and φ were selected as 10 Ω,500 μH, and 0.4695 V, respectively. The resonant frequency, denoted as ωo=2 π60 rads, was maintained by choosing

$C = {\frac{1}{\omega_{o}^{2}L}.}$

FIG. 8 is a circuit diagram of an exemplary N oscillators. Microgridscan be made up of N inverters connected across a load. Each oscillatoris connected to a common node through a branch impedance, z net (s),which may contain any combination of linear circuit elements. Thevoltage at the common node to which all the network impedances areconnected is denoted by v_(load). When no load is connected to thecommon node, the oscillator synchronization condition can be shown to beindependent of the number of oscillators. The case where the oscillatorsdeliver power to a load, e.g., connected at the common node can beanalyzed and it can be demonstrated that the synchronization conditionis the same as the case with no load. This implies that the system canbe designed independent of the load parameters and knowledge of numberof inverters.

With the system of oscillators connected to no load, the j^(th)oscillator output current is given by:

$\begin{matrix}{{i_{j}(s)} = {\frac{1}{z_{net}(s)}{\left( {{v_{j}(s)} - {v_{load}(s)}} \right).}}} & (48)\end{matrix}$

Since the output currents sum to zero,

$\begin{matrix}{0 = {{\sum\limits_{k = 1}^{N}\;{i_{k}(s)}} = {\frac{1}{z_{net}(s)}{\left( {\left( {\sum\limits_{k = 1}^{N}\;{v_{k}(s)}} \right) - {{Nv}_{load}(s)}} \right).}}}} & (49)\end{matrix}$

Rearranging terms,

$\begin{matrix}{{v_{load}(s)} = {\frac{1}{N}{\sum\limits_{k = 1}^{N}\;{{v_{k}(s)}.}}}} & (50)\end{matrix}$

Substituting (50) in (48) leads to

$\begin{matrix}{{i_{j}(s)} = {\frac{1}{z_{net}(s)}\left( {{v_{j}(s)} - {\frac{1}{N}{\sum\limits_{k = 1}^{N}\;{v_{k}(s)}}}} \right)}} & (51)\end{matrix}$

Writing all output currents in matrix form gives:

$\begin{matrix}{{{i(s)} = {{\frac{1}{z_{net}(s)}\left( {I_{N} - {\frac{1}{N}11^{T}}} \right){v(s)}} = {\frac{1}{{Nz}_{net}(s)}\Gamma\;{v(s)}}}},{where}} & (52) \\{{\Gamma = {{{NI}_{N} - 11^{T}} = \begin{bmatrix}{N - 1} & {- 1} & \ldots & {- 1} \\{- 1} & {N - 1} & \ldots & {- 1} \\\vdots & \vdots & \ddots & \vdots \\{- 1} & {- 1} & \ldots & {N - 1}\end{bmatrix}}},} & (53)\end{matrix}$

for this particular network. The smallest non-zero eigenvalue, λ₂, ofthis Laplacian is equal to N. Comparing (52) with (9),

$\begin{matrix}{{Y(s)} = {\frac{1}{{Nz}_{net}(s)}\Gamma}} & (54)\end{matrix}$

Furthermore, by referring to (10) and (30), for the no-load case

$\begin{matrix}\left\{ \begin{matrix}{{\alpha(s)} = 0} \\{{\beta(s)} = \left( {{Nz}_{net}(s)} \right)^{- 1}} \\{{\zeta(s)} = {z_{osc}(s)}}\end{matrix} \right. & (55)\end{matrix}$

Applying ζ(s), β(s), and λ₂ for this network in the linear fractionaltransformation of Theorem 1 leads to

$\begin{matrix}\begin{matrix}{{F\left( {{\zeta(s)},{{\beta(s)}\lambda_{2}}} \right)} = {\left( {1 + {{\zeta(s)}{\beta(s)}\lambda_{2}}} \right)^{- 1}{\zeta(s)}}} \\{= \frac{z_{osc}(s)}{1 + {{z_{osc}(s)}\left( {{Nz}_{net}(s)} \right)^{- 1}N}}} \\{= \frac{{z_{osc}(s)}{z_{net}(s)}}{{z_{net}(s)} + {z_{osc}(s)}}}\end{matrix} & (56)\end{matrix}$

F(ζ(s), β(s)λ₂) equals the impedance of the parallel combination ofz_(osc)(s) and z_(net)(s). Applying (29) of Theorem 1 gives thefollowing synchronization condition:

$\begin{matrix}{{\sup\limits_{\omega \in \bullet}\mspace{14mu}{\frac{{z_{net}\left( {j\;\omega} \right)}{z_{osc}\left( {j\;\omega} \right)}}{{z_{net}\left( {j\;\omega} \right)} + {z_{osc}\left( {j\;\omega} \right)}}}_{2}\sigma} < 1.} & (57)\end{matrix}$

Note that the condition for synchronization is independent of N anddepends only on the impedance of the oscillator linear subsystem,z_(osc)(s), and the branch impedance, z_(net)(s).

FIG. 9 is a circuit diagram of an exemplary N oscillators with a linearload. The system of oscillators are connected to a passive linear load.The load can be made up of any arbitrary combination of passive LTIcircuit elements. The j^(th) oscillator output current is given by (48),and further,

$\begin{matrix}{{v_{load}(s)} = {{z_{load}(s)}{\sum\limits_{k = 1}^{N}\;{{i_{k}(s)}.}}}} & (58)\end{matrix}$

Substituting (58) in (48) yields

$\begin{matrix}{{{i_{j}(s)} = {\frac{1}{z_{net}(s)}\left( {{v_{j}(s)} - {{z_{load}(s)}{\sum\limits_{k = 1}^{N}\;{i_{k}(s)}}}} \right)}},} & (59)\end{matrix}$

from which v_(j) (s) can be isolated to get

$\begin{matrix}{{v_{j}(s)} = {{{z_{net}(s)}{i_{j}(s)}} + {{z_{load}(s)}{\sum\limits_{k = 1}^{N}\;{{i_{k}(s)}.}}}}} & (60)\end{matrix}$

Collecting all terminal voltages in matrix form yieldsv(s)=(z _(net)(s)I _(N) +z _(load)(s)11^(T))i(s).  (61)

Comparing (61) with (9) indicatesY ⁻¹(s)=z _(net)(s)I _(N) +z _(load)(s)11^(T).  (62)

To invert (62), begin by diagonalizing 1 1^(T)=Q

Q^(T), where Λ={0, . . . , 0, N}ε□^(N×N), to get

$\begin{matrix}\begin{matrix}{{Y^{- 1}(s)} = {{{z_{net}(s)}I_{N}} + {{z_{load}(s)}Q\;\Lambda\; Q^{T}}}} \\{= {{z_{net}(s)}{Q\left( {I_{N} + {\frac{z_{load}(s)}{z_{net}(s)}\Lambda}} \right)}{Q^{T}.}}}\end{matrix} & (63)\end{matrix}$

It can be useful to determinez _(eq)(s):=z _(net)(s)+Nz _(load)(s).  (64)

Inverting the expression in (63) yields

$\begin{matrix}\begin{matrix}{{Y(s)} = {\frac{1}{z_{net}(s)}{Q\left( {I_{N} - {\frac{z_{load}(s)}{z_{eq}(s)}\Lambda}} \right)}Q^{T}}} \\{= {\frac{1}{{z_{net}(s)}{z_{eq}(s)}}{Q\left( {{{z_{eq}(s)}I_{N}} - {{z_{load}(s)}\Lambda}} \right)}Q^{T}}} \\{= {\frac{1}{{z_{net}(s)}{z_{eq}(s)}}\left( {{\left( {{z_{net}(s)} + {{Nz}_{load}(s)}} \right)I_{N}} - {{z_{load}(s)}11^{T}}} \right)}} \\{= {\frac{1}{{z_{net}(s)}{z_{eq}(s)}}\left( {{{z_{net}(s)}I_{N}} + {{z_{load}(s)}\left( {{NI}_{N} - 11^{T}} \right)}} \right)}} \\{{= {\frac{1}{{z_{net}(s)}{z_{eq}(s)}}\left( {{{z_{net}(s)}I_{N}} + {{z_{load}(s)}\Gamma}} \right)}},}\end{matrix} & (65)\end{matrix}$

where in the third line above, the definition of z_(eq) (s) from (64)was used, and in the last line, the Γ determined in (53) was utilized.Comparing (65) with (10), and using (30), it is evident that for thelinear-load case:

$\begin{matrix}\left\{ \begin{matrix}{{\alpha(s)} = {z_{eq}^{- 1}(s)}} \\{{\beta(s)} = {{z_{load}(s)}{z_{net}(s)}{z_{eq}^{- 1}(s)}}} \\{{\zeta(s)} = {{z_{osc}(s)}{z_{eq}(s)}\left( {{z_{osc}(s)} + {z_{eq}(s)}} \right)^{- 1}}}\end{matrix} \right. & (66)\end{matrix}$

As the system synchronizes and the interaction between oscillatorsdecays to zero, the effective impedance observed from the output of theoscillator is equal to z_(eq) (s)=z_(net)(s)+N z_(load)(s). In otherwords, the effective load seen by each oscillator during synchronizedsystem conditions is equal to z_(eq) (s).

For the ζ(s) and β(s) in (66), it follows that

$\begin{matrix}\begin{matrix}{{F\left( {{\zeta(s)},{{\beta(s)}\lambda_{2}}} \right)} = {\left( {1 + {{\zeta(s)}{\beta(s)}\lambda_{2}}} \right)^{- 1}{\zeta(s)}}} \\{= \frac{z_{osc}{z_{eq}\left( {z_{osc} + z_{eq}} \right)}^{- 1}}{1 + {z_{osc}{z_{eq}\left( {z_{osc} + z_{eq}} \right)}z_{load}z_{net}^{- 1}z_{eq}^{- 1}N}}} \\{= \frac{z_{osc}}{{z_{eq}^{- 1}\left( {z_{osc} + z_{eq}} \right)} + {z_{osc}z_{eq}^{- 1}z_{load}z_{net}^{- 1}N}}} \\{= \frac{z_{osc}}{1 + {z_{osc}{z_{eq}^{- 1}\left( {1 + {z_{load}z_{net}^{- 1}N}} \right)}}}} \\{= {\frac{z_{osc}}{1 + {z_{osc}{z_{eq}^{- 1}\left( {z_{eq}z_{net}^{- 1}} \right)}}} = {\frac{{z_{osc}(s)}{z_{net}(s)}}{{z_{net}(s)} + {z_{osc}(s)}}.}}}\end{matrix} & (67)\end{matrix}$

Applying (29), the synchronization condition

$\begin{matrix}{{\sup\limits_{\omega \in \bullet}\mspace{14mu}{\frac{{z_{net}\left( {j\;\omega} \right)}{z_{osc}\left( {j\;\omega} \right)}}{{z_{net}\left( {j\;\omega} \right)} + {z_{osc}\left( {j\;\omega} \right)}}}_{2}\sigma} < 1} & (68)\end{matrix}$

follows, which is the same condition as the no-load case in (57). Thesynchronization condition is independent of the number of oscillatorsand the load impedance.

FIG. 10 is a plot of exemplary inverter output currents, voltages, andvoltage synchronization error. Simulation results validate thesynchronization condition. It can be demonstrated that a system ofinverters controlled as deadzone oscillators satisfying (29) cansynchronize in a passive electrical network and deliver power to a load.In the case studies, a network with the topology in FIG. 9 can beconsidered. The network branch impedance is given byz_(net)(s)=sL_(net)+R_(net), where L_(net) and R_(net) correspond to thecombined line and inverter-output-filter inductance and resistance,respectively, e.g., the inverter output-filter inductance is used toreduce harmonics in the inverter output current that arises due toswitching). It is assumed that the load is resistive such thatz_(load)(s)=R_(load).

TABLE 1 System parameters used in the case studies. Case Study I, II N100    R 8.66 Ω L 433.2 μH C 16.2 mF σ 1.15 S φ 146.1 V ε  0.170 R_(net)0.1 Ω, 0.02 Ω L_(net)   500 μH R_(load) 91.96 mΩ

For this system, the linear fractional transformation is given by

$\begin{matrix}\begin{matrix}{{F\left( {{z_{osc}(s)},\;{z_{net}^{- 1}(s)}} \right)} = \frac{{z_{osc}(s)}{z_{net}(s)}}{{z_{net}(s)} + {z_{osc}(s)}}} \\{= \frac{\frac{1}{C}s}{s^{2} + {\frac{1}{RC}s} + \frac{1}{LC} + {\frac{s}{C}\left( {{L_{net}s} + R_{net}} \right)^{- 1}}}} \\{= {\frac{\left( {{L_{net}s} + R_{net}} \right)s}{\begin{matrix}{{L_{net}{Cs}^{3}} + {\left( {\frac{L_{net}}{R} + {R_{net}C}} \right)s^{2}} +} \\{{\left( {\frac{L_{net}}{L} + \frac{R_{net}}{R} + 1} \right)s} + \frac{R_{net}}{L}}\end{matrix}}.}}\end{matrix} & (69)\end{matrix}$

The design objective is to select R, L, C, σ, and φ for a givenz_(net)(s), such that the load voltage and system frequency meetsperformance specifications. To help guarantee synchronization, thesystem design satisfies the synchronization condition∥F(z_(osc)(jω),z_(net) ⁻¹(jω)∥_(∞)σ<1. The material below describes aparameter selection technique which determines that the invertersoscillate at the desired frequency and that in steady-state v_(load)stays within ±5% of the rated voltage across the entire load range(no-load to maximum rated load). In case studies I and II, a powersystem including 100 parallel inverters which are each rated for 10 kWis simulated. The RMS voltage and frequency ratings of the system are220 V and 60 Hz, respectively, and the maximum load power is 1 MW. Thesystem parameters used in each case study are summarized in Table 1.

In FIG. 10, inverter output currents, voltages, and voltagesynchronization error is shown in the case when ∥F(z_(osc)(jω),z_(net)⁻¹(jω)∥_(∞)σ<1. Substituting the corresponding values in Table 1 into(69), ∥F(z_(osc)(jω),z_(net) ⁻¹(jω)∥_(∞)σ=0.77<1. Therefore,synchronization of the oscillator system is guaranteed. At t=0, currentsare zero and the oscillator capacitor voltages are chosen to beuniformly distributed between ±10 V. Initially, the system contains noload. After successful synchronization, the load is abruptly added att=300 ms. As shown in FIG. 8, the voltage stays within ±5% of the ratedvalue during steady-state conditions.

FIG. 11 is a plot of an exemplary evolution of state-variables duringstartup in the presence of a load. Waveforms for 10 (out of 100simulated) are shown for clarity. A second simulation was prepared todemonstrate synchronization in the presence of the load. The load isconnected at t=0 s. Given initial conditions as used above, FIG. 11shows the trajectories of the state-variables (only 10 out of 100waveforms are shown for clarity). The inductor current within theoscillator RLC circuit is denoted as i_(L). The state-variables reach astable limit-cycle.

FIG. 12 is a plot of exemplary inverter output currents, voltages, andvoltage synchronization error when ∥F(z_(osc)(jω),z_(net)⁻¹(jω)∥_(∞)σ=2.78 {hacek over (Z)}1 and synchronization is notguaranteed. All parameters, except R_(net), and initial conditionsdescribed in FIG. 11 were re-used. The value of R_(net) was reduced suchthat ∥F(z_(osc)(jω),z_(net) ⁻¹(jω)∥_(∞)σ≧1, and synchronization is notguaranteed. At t=0, the load is connected to the system. As illustratedin FIG. 12, the inverters do not reach synchrony.

The inverters in a microgrid can be controlled to act as nonlinearoscillators. The resulting microgrid is modular and does not requirecommunication between inverters. A general theorem gives asynchronization condition for N nonlinear oscillators coupled through anLTI electrical network. When this theorem is applied to N oscillatorsconnected in parallel across a load, the synchronization condition isindependent of N and the load parameters. Simulation results are used tosubstantiate the analytical framework and illustrate the merit of theapplication. Practical design and implementation techniques aredescribed below.

FIG. 13A is a circuit diagram of exemplary inverters that are controlledto emulate the exemplary oscillators in FIG. 13B. The parallel inverterscan be coordinated such that each inverter is digitally controlled,e.g., virtual oscillator control, to mimic the nonlinear oscillatorsdiscussed above. In one implementation, a microgrid with N parallelinverters can be controlled as deadzone oscillators. The systemsynchronization condition described above depends only on the oscillatorparameters and output filter impedance of a single inverter. Here, theresult is applied by putting forward a design procedure based on theparameters of one inverter. A system and method for adding invertersinto an energized microgrid is also introduced and exemplary results arepresented.

A system of parallel single-phase voltage source inverters in amicrogrid is shown in FIG. 13A. Objectives are to control the system ofinverters such that communication between inverter controllers isunnecessary, the load is shared equally between inverters, allalternating current (AC) outputs synchronize and oscillate at thedesired frequency, inverter synchronization is guaranteed for any numberof parallel inverters, and load voltage is maintained within desiredlimits. The inverters in FIG. 13A can be controlled to mimic the systemof parallel deadzone oscillators in FIG. 13B such that the aboveobjectives are satisfied. A method of controlling inverters by digitalmeans such that they act as deadzone oscillators is also described.

FIG. 14 is a circuit diagram of an exemplary single-phase voltage sourceinverter with virtual oscillator control. To control a single-phaseinverter such that it mimics a deadzone oscillator, the differentialequations of the oscillator can be programmed on the digital controllerof the inverter. Because the deadzone oscillator need not physicallyexist, it is described as being virtual. A representative implementationof the control on a single-phase H-bridge inverter is given in FIG. 14.The measured output current of the inverter is scaled by κ₁ andextracted from the virtual oscillator. The oscillator voltage is thenmultiplied by κ_(v) and used to generate a modulation signal, m. Theinverter switching signals are generated by applying a pulse widthmodulation technique, e.g., sine-triangle pulse width modulation oranother technique.

The inverter emulates the dynamics of the nonlinear oscillator such thatthe inverter output voltage, v, follows the scaled oscillator voltage,κ_(v)v_(osc). The current extracted from the virtual oscillator is equalto the scaled output current, κ_(i)i. The scaling parameters, κ_(v) andκ_(i), can be used to aid the design process, as described below.

The inverter is an electronics device or circuitry that changes directcurrent (DC) to alternating current (AC). The input voltage, outputvoltage and frequency, and overall power handling, can be implementationdependent. For exemplary purposes, an inverter 1500 includes powerelectronics 1502. The power electronics 1502 includes a DC energy source1504 which generates a DC voltage V_(dc), e.g., by battery, fuel cell,etc., and switches 1506 to convert the DC voltage V_(dc) to an ACvoltage V 1508. FIGS. 34-37 illustrate an exemplary circuit for threephase voltage. The switches 1506 can be implemented with semiconductors,etc. Z_(net) 1510 is filtering, e.g., passive inductors and capacitors,connected at an output of the of the power electronics 1502 to filternoise from the power electronics switching. At 1512, the current i isdetermined, e.g., measure or otherwise captured, and digitized, e.g.,with an analog-to-digital converter.

The digitized, measured current i can be inputted to microcontroller1520 using analog to digital converters. The measured current i can bescaled by value K_(i) 1522 to obtain K_(i)i for inputting into thevirtual oscillator 1524. The virtual oscillator 1524 can be implementedwith physically and/or with code stored in memory and executed by aprocessor of the microcontroller 1520. The virtual oscillator 1524 canbe emulated in real time and the voltage across the oscillator v_(osc)1525 is scaled by K_(v) 1526, and divided by the value of the dc-linkvoltage 1528, and inputted as m into the pulse width modulator (PWM)1530 for sending signals to control the switching of the switches 1506.The average voltage, V, across the terminals of the power electronics1502 follows the scaled oscillator voltage K_(v)V_(osc) after 1526. Asdescribed in more detail, below, the virtual oscillator 1524 and itsparameters, R, L, C, σ, and φ, can be selected for a given z net (s),such that the load voltage and system frequency meets performancespecifications while also to guaranteeing synchronization of inverters.The system design can satisfy the synchronization condition∥F(z_(osc)(jω),z_(net) ⁻¹(jω)∥_(∞)σ<1, e.g., if the synchronizationcondition is less than one then inverters will synchronize. The virtualoscillator 1524 can be designed for one inverter and when the sameinverters are added to the system the AC output of the inverters canself-synchronize such that the inverter ac terminal voltages differencesdecay to zero.

When the inverters are connected in a network, e.g., power grid or othernetwork, the inverters utilize measurements local to themselves, withouta need to exchange information between the inverters, yet the invertersoscillate in unison. Voltage limits are respected across a no-load to amaximum rated load range. The inverters can automatically synchronizeacross network, e.g., upon adding or removing a power electronicsinverter to the network. The synchronization can occur without a phaselock loop to generate sinusoidal reference waveforms. Thesynchronization can also occur without a proportional-integral (PI) orproportional-integral-derivative (PID) controller. The synchronizationis agnostic to the number of inverters and loads. Since the controlleracts on the instantaneous measurements and does not require real andreactive power calculations, the controller can be significantly faster.The inverters can share a load power proportional to their size, e.g.,the inverters can provide as much power to the load in accordance withtheir ratings. For example, an inverter that has twice the power ratingas another inverter can automatically provide twice as much power as thesmaller inverter, without the need for communicating the power ratinginformation between inverters.

The inverters 1500 can further include a pre-synchronization circuit,described below in FIG. 19, to match the inverters before adding them tothe network. The pre-synchronization circuit can prevent excessivecurrent through the inverter 1500 when adding the inverter 1500 to thesystem. The pre-synchronization circuit processes the measured voltageacross the output terminals of the power electronics switches 1502 suchthat the oscillator voltage locks on to the voltage at the terminals.Once the internal oscillator voltage has converged to the measuredvoltage, the inverter 1500 is energized and delivers power to thenetwork. The power electronics switches 1502 can also be interleaved tohelp cancel distortion generated by the system of inverters.

FIG. 15 is a circuit model of N connected inverters and associatedvirtual oscillator controllers coupled through a microgrid network. Thedynamic system equations of an inverter system with the controlimplementation are described below. The parallel topology with a load isconsidered since it is more general than the system with no load. Thevirtual system of oscillators reside within the digital controllers, anda system of inverters is coupled through a physical electrical network.After the system equations are derived, the theorem described above canbe used to attain a synchronization condition.

Each inverter is digitally controlled to mimic the dynamics of adeadzone oscillator. A system of N connected inverters with virtualoscillator control can be modeled using the diagram in FIG. 15. Thesystem on the left represents the N virtual oscillators which residewithin the microcontroller associated for each inverter. Dynamics of thecircuit, e.g., equations and transfer functions of the systems andmethods, can be programmed into the microcontroller. The right-hand sideof the diagram is a representation of the N inverters and the microgridnetwork. The averaged dynamics of the j^(th) inverter is modeled as thecontrolled voltage source v_(j) and v=[v₁(s), . . . , v_(N) (s)]^(T) isthe vector of inverter voltages. Using FIG. 15, the inverter voltagesand currents are related byi(s)=Y(s)v(s),  (70)

where i(s)=[i₁(s), . . . , i_(N)]^(T) of is the vector of inverteroutput currents and Y(s) is the network admittance transfer matrix. Theadmittance matrix for the network in FIG. 15 can be written as

$\begin{matrix}{{Y(s)} = {\frac{1}{{z_{net}(s)}{z_{eq}(s)}}{\left( {{{z_{net}(s)}I_{N}} + {{z_{load}(s)}\Gamma}} \right).}}} & (71)\end{matrix}$

Recall that z_(eq) (s) is determined asz _(eq)(s):=z _(net)(s)+Nz _(load)(s),  (72)

and the Laplacian, Γ, for this network is given by (53).

From FIG. 15, it follows that the voltage of the j^(th) virtualoscillator, v_(oscj) (s), can be expressed asv _(oscj)(s)=z _(osc)(s)(i _(srcj)(s)−i _(oscj)(s)),∀j=1, . . .,N,  (73)

where i_(oscj)(s) is the output current the j^(th) virtual oscillator.The output voltages and currents of the j^(th) oscillator and inverterare related byv _(oscj)(s)κ_(v) =v _(j)(s),  (74)andi _(oscj)(s)=i _(j)(s)κ_(i);  (75)

where κ_(v), κ₁ε

are the voltage and current scaling gains, respectively. From Equations(74) and (75), it is apparent that the inverter voltage equals thescaled oscillator voltage and the oscillator output current is thescaled inverter output current. It can be useful to define:κ:=κ_(i)κ_(v)  (76)

Writing all N oscillator voltages in matrix form gives

$\begin{matrix}\begin{matrix}{{v_{osc}(s)} = {{Z_{osc}(s)}\left( {{i_{src}(s)} - {i_{osc}(s)}} \right)}} \\{= {{Z_{osc}(s)}\left( {{i_{src}(s)} - {\kappa_{i}{i(s)}}} \right)}} \\{= {{{Z_{osc}(s)}{i_{src}(s)}} - {\kappa\;{Z_{osc}(s)}{Y(s)}{{v_{osc}(s)}.}}}}\end{matrix} & (77)\end{matrix}$

In the second line, (75) is substituted for i_(osc)(s), and in the lastline i(s)=Y(s)v(s)=κ_(v)Y(s) v_(osc)(s) comes from the substitution of(74) into v(s). Solving for v_(osc)(s) in (77) yields:

$\begin{matrix}\begin{matrix}{{v_{osc}(s)} = {\left( {I_{N} + {\kappa\;{Z_{osc}(s)}{Y(s)}}} \right)^{- 1}{Z_{osc}(s)}{i_{src}(s)}}} \\{{= {{F\left( {{Z_{osc}(s)},{\kappa\;{Y(s)}}} \right)}{i_{src}(s)}}},}\end{matrix} & (78)\end{matrix}$

where F(Z_(osc)(s),κY(s)) is the linear fractional transformation asdetermined in (14). Using (78), the system of coupled virtualoscillators can be represented as the block-diagram in FIG. 16 where thelinear and nonlinear portions of the system are compartmentalized byF(Z_(osc)(s),κY (s)) and g(v)=[g(v₁), . . . , g(v_(N))]^(T),respectively. Linear and nonlinear portions of the system are containedin F(Z_(osc)(s),κY(s)) and g(v)=[g(v₁), . . . , g(v_(N))]^(T),respectively.

To analyze global asymptotic synchronization in the network of virtualoscillators described above, synchronization can be described by thecondition

$\begin{matrix}{{{{\lim\limits_{t\rightarrow\infty}\mspace{14mu}{v_{oscj}(t)}} - {v_{osck}(t)}} = {0\mspace{14mu}{\forall j}}},{k = 1},\ldots\mspace{14mu},{N.}} & (79)\end{matrix}$

Because the inverter voltages equal the scaled oscillator voltageswithin the controllers, it follows that virtual oscillatorsynchronization implies inverter voltage synchronization. Applying theprojector matrix to the vector of oscillator voltages gives

$\begin{matrix}\begin{matrix}{{{{\overset{\sim}{v}}_{osc}(t)}^{T}{{\overset{\sim}{v}}_{osc}(t)}} = {\left( {\prod{v_{osc}(t)}} \right)^{T}\left( {\prod{v_{osc}(t)}} \right)}} \\{= {\frac{1}{2\; N}{\sum\limits_{j = 1}^{N}\;{\sum\limits_{k = 1}^{N}\;{\left( {{v_{oscj}(t)} - {v_{osck}(t)}} \right)^{2}.}}}}}\end{matrix} & (80)\end{matrix}$

Recall that the projector matrix, Π, is determined in (6). Theoscillator voltage synchronization results when {tilde over(v)}_(osc)(t)=Πv_(osc)(t)→0 as t→∞. Following along the same lines ofthe previous analysis, it can be shown that

$\begin{matrix}\begin{matrix}{{{\overset{\sim}{v}}_{osc}(s)} = {\left( {I_{N} + {\kappa\;{Z_{osc}(s)}{Y(s)}}} \right)^{- 1}{Z_{osc}(s)}{{\overset{\sim}{i}}_{src}(s)}}} \\{= {{F\left( {{Z_{osc}(s)},{\kappa\;{Y(s)}}} \right)}{{{\overset{\sim}{i}}_{src}(s)}.}}}\end{matrix} & (81)\end{matrix}$

The map {tilde over (g)}:

^(N)→

^(N) captures the impact of g(v) in the corresponding differentialsystem and is determined as follows:{tilde over (g)}:{tilde over (v)} _(osc) →ĩ _(src)  (82)

Equations (81) and (82) form a description of the dynamics in thecorresponding differential system. Furthermore, the results permit theblock-diagram representation of the differential system in FIG. 17,where, as in FIG. 16, the linear and nonlinear subsystems arecompartmentalized using F(•,•) and {tilde over (g)}, respectively.

A sufficient synchronization condition for the inverter system in FIG.15 is given. Corollary 1 is the network of N inverters in FIG. 15synchronizes in the sense of (79), if

$\begin{matrix}{{\max\limits_{\omega \in \bullet}\mspace{14mu}{{\frac{\kappa^{- 1}{z_{net}\left( {j\;\omega} \right)}{z_{osc}\left( {j\;\omega} \right)}}{{\kappa^{- 1}{z_{net}\left( {j\;\omega} \right)}} + {z_{osc}\left( {j\;\omega} \right)}}}_{2}\sigma}} < 1.} & (83)\end{matrix}$

The synchronization condition in (83) is similar to the condition givenin (68). A difference is that the impedance, z_(net), is scaled by κ⁻¹.As before, the synchronization condition is independent of the number ofinverters and the load parameters.

A system and method for control design is described. A set of guidelinesfor parameter selection is also presented. In addition, a techniquewhich facilitates the addition of inverters in an energized system isdescribed. Because the synchronization condition is independent of N andthe load parameters, for explanation purposes the task of system designis reduced to that of one inverter and its associated control. Aninverter is provided which has a given filter impedance, z net, andpower rating, P_(max). Furthermore, a system frequency rating, ω_(o), isgiven. The peak load voltage is allowed to deviate between upper andlower limits, v_(max) and v_(min), respectively. Also, v_(pk) denotesthe peak value of v load in steady-state conditions.

The virtual oscillator parameters R, L, C, σ, φ, κ_(v), and κ_(i) areselected such that

$\begin{matrix}\left\{ \begin{matrix}{\frac{1}{\sqrt{LC}} = \omega_{o}} \\{\sigma > \frac{1}{R}} \\{v_{pk} = {v_{\max}\mspace{14mu}{under}\mspace{14mu}{no}\text{-}{load}\mspace{14mu}{conditions}}} \\{{v_{pk} = {v_{\min}\mspace{14mu}{undermaximumratedloadconditions}}},P_{\max}} \\{\overset{\backprime}{o} = {\sqrt{\frac{L}{C}}\left( {\sigma - \frac{1}{R}} \right)\mspace{14mu}{is}\mspace{14mu}{minimized}}} \\{{\sup\limits_{\omega \in \bullet}\mspace{14mu}{\frac{\kappa^{- 1}{z_{net}\left( {j\;\omega} \right)}{z_{osc}\left( {j\;\omega} \right)}}{{\kappa^{- 1}{z_{net}\left( {j\;\omega} \right)}} + {z_{osc}\left( {j\;\omega} \right)}}}_{2}\sigma} < 1}\end{matrix} \right. & (84)\end{matrix}$

Stated another way, the objectives ensure: a) the inverters oscillate atω_(o). b) voltage limits are respected such that v_(min)≦v_(pk)≦v_(max)across the entire load range (no-load to maximum rated load). C) thedistortion on the sinusoidal output is reduced, and d) thesynchronization condition is satisfied.

FIG. 18 is a circuit model used in control parameter selection. Giventhe nonlinear nature of the system and lack of analytical tools for thedeadzone oscillator, an iterative design process is shown. In steps 3and 4, the analyst simulate the model until steady-state conditions arereached. In step 2, the parameters R and C are selected such that theindividual inverter has a stable oscillation at the rated systemfrequency. The constraint

$\sigma > \frac{1}{R}$ensures that a stable oscillation exists, and the relation

$\omega_{o} = \frac{1}{\sqrt{LC}}$guarantees that the circuit oscillates at the rated frequency, ω_(o).Steps 3 and 4, which utilize a time-domain simulation of the model inFIG. 18, are motivated by the amplitude of v load decreasing as the realpower consumed in the load increases. Consequently, the maximum andminimum load voltages correspond to the no load and full-rated loadcases, respectively. Once steps 3 and 4 are complete, thenv_(min)≦v_(pk)≦v_(max) can be satisfied across the entire rated loadrange. If the synchronization condition is satisfied, a system of Ninverters with identical design parameters synchronizes.

FIG. 19 is a circuit diagram of an exemplary inverter controller with apre-synchronization circuit. A condition for global asymptoticsynchronization is independent of N and the load parameters. As thenumber of inverters changes, the system can synchronize. If thesynchronization condition is satisfied, inverters can be added andremoved as needed and the system synchronizes in the steady-state.Despite this favorable property, it is possible that system transientscan be undesirably large when inverters are added. To facilitate aseamless addition of inverter units and avoid hardware damage due tolarge transients, a pre-synchronization circuit is added.

For t<t_(o), there are N inverters operating in a microgrid with a load.At t=t_(o), an additional inverter or multiple inverters are added tothe system. The additional inverter is capable of measuring the commonnode voltage, v_(load), prior to being added. As shown in FIG. 19, fort<t_(o) the virtual oscillator of the inverter to be added can beaugmented with a pre-synchronization circuit which includes: i) a scaledfilter impedance, κ⁻¹ z_(net), ii) a virtual load, z_(virt), and iii) avoltage source which follows v_(load) and is interfaced through a seriesresistor, R_(link). Before the additional inverter is connected to thesystem, it is assumed that the N operational inverters are synchronizedand in steady-state. A purpose of the pre-synchronization circuit is tobring the state-variables of the additional inverter controller as closeas possible to synchronization with the operational inverter controllersbefore being added to the system.

The pre-synchronization control in FIG. 19 is allowed to reachsteady-state before the inverter is added to the system at t=t_(o).During steady-state conditions: i) if the virtual load is chosen suchthat z_(virt)≈κ⁻¹Nz_(load), then the virtual oscillator state variablesin the additional unit closely match that of the operational invertercontrollers, and ii) v_(virt)≈κ_(v) ⁻¹v_(load), where v_(virt) is thevoltage across the virtual load and v_(load) is load voltage in theenergized system. During these conditions, inverters can be added withminimal system transients. When the inverter is added at t=t_(o), thepre-synchronization circuit is removed and the original virtualoscillator control remains.

FIG. 20 is a circuit diagram of an exemplary inverter system. Threesingle-phase H-bridge inverters and accompanying control can beconfigured to deliver power to a load. The inverters are interfaced to acommon node through branch impedances z_(net)(s)=sL_(f)+R_(f). Thefilter inductance is denoted as L_(f) and the combined resistance of theinductor windings, connectors, and conductors is lumped into R_(f). Forexplanation purposes, the inverters are rated to deliver approximately50 W and has a 100 V dc source at the input. For illustrative purposes,the controllers were implemented on a Texas Instruments TMS320F28335microcontroller and each inverter can utilize an independent controlloop.

TABLE 2 Exemplary control and hardware parameters: System parameters ω₀= 2 π 60 ^(rad) σ = 1 S ^(v)rated = 60 {square root over (2V)} R = 10 Ω^(v)max = 1.05 ^(v)rated L = 500 μH ^(v)min = 0.95 ^(v)rated$C = {\frac{1}{L\;\omega_{o}^{2}} \approx {14.07\mspace{14mu}{mF}}}$K_(v) = 60{square root over (2)} φ = 0.4695 V K_(i) = 0.1125 ε = 0.170^(z)virt = 2 κ⁻¹ 57.5 Ω R_(f) = 1 Ω ^(R)link = κ⁻¹ 100 Ω L_(f) = 6 mH

For the system structure in FIG. 20, the linear fractionaltransformation is given by

$\begin{matrix}\begin{matrix}{{F\left( {{z_{osc}(s)},{\kappa\;{z_{net}^{- 1}(s)}}} \right)} = \frac{\kappa^{- 1}{z_{net}\left( {j\;\omega} \right)}{z_{osc}\left( {j\;\omega} \right)}}{{\kappa^{- 1}{z_{net}\left( {j\;\omega} \right)}} + {z_{osc}\left( {j\;\omega} \right)}}} \\{= \frac{\frac{1}{C}s}{s^{2} + {\frac{1}{RC}s} + \frac{1}{LC} + {\frac{s}{C}{\kappa\left( {{L_{f}s} + R_{f}} \right)}^{- 1}}}} \\{= {\frac{\left( {{L_{net}s} + R_{net}} \right)s}{\begin{matrix}{{L_{net}{Cs}^{3}} + {\left( {\frac{L_{net}}{R} + {R_{net}C}} \right)s^{2}} +} \\{{\left( {\frac{L_{net}}{L} + \frac{R_{net}}{R} + \kappa} \right)s} + \frac{R_{net}}{L}}\end{matrix}}.}}\end{matrix} & (85)\end{matrix}$

The peak voltage and frequency ratings of the system are 60√{square rootover (2)} V and 60 Hz, respectively. The parameters in Table 2 wereselected such that the load voltage stayed within ±5% of the rated valueacross the load range (no-load to maximum rated load). Substituting thecorresponding values in Table 2 into (85), ∥F(z_(osc)(jω),κz_(net)⁻¹(jω)∥_(∞)σ=0.93<1. Therefore, synchronization of the inverter systemis guaranteed. The inverter system can be connected to a variety ofloads. The synchronization condition can be used for passive linearloads and mechanical loads. System startup and load transients can beconsidered in addition to inverter removal and addition dynamics.

FIG. 21 is an oscilloscope screenshot of exemplary measured inverteroutput currents and load voltage during system startup with a resistiveload. In the oscilloscope screenshots, the top three waveforms cancorrespond to the measured inverter currents and the bottom waveform isthe load voltage. To mimic non-ideal startup conditions, the virtualoscillator voltages were initialized such that κ_(v) v_(osc)(0)=v(0)=[5V, 4V, 3V]^(T) for each of the respective invertercontrollers. FIG. 21 shows the system currents and load voltages duringstart-up in the presence of a 50Ω load. In this example, R_(load)undergoes step changes between 500Ω and 71.4Ω.

FIGS. 22(a) and 22(b) are oscilloscope screenshots of resistive loadtransients showing exemplary inverter output currents and load voltageduring (a) load step-up and (b) load step-down. The inverters increaseand decrease their output currents almost instantaneously as the loadpower changes. Furthermore, the load voltage amplitude remains nearlyconstant during transients.

FIGS. 23(a) and 23(b) are oscilloscope screenshots of inverter outputcurrents and load voltage when an inverter is (a) removed and (b) addedin the presence of a resistive load. In the case when the number ofinverters in the microgrid undergoes a change, system dynamics duringinverter removal and addition are shown in FIGS. 23(a) and 23(b),respectively. In both transients, R_(load)=66Ω. During inverter removal,the remaining units quickly compensate by increasing their outputcurrents. The pre-synchronization technique, with parameters assummarized in Table 2 is implemented before adding inverter #3 back intothe system. As demonstrated in FIG. 23(b), system transients arerelatively small during unit addition. Furthermore, the load voltagewaveform is largely unaffected during both inverter removal and additiontransients.

FIGS. 24(a) and 24(b) are circuit diagrams of exemplary circuits for (a)linear RLC load and (b) nonlinear diode bridge rectifier load. Systemstart-up is demonstrated when the switch in the RLC load is closed. Thevirtual oscillators associated with each inverter were initialized suchthat v(0)=[5 V, 4 V, 3 V]^(T).

FIG. 25 is an oscilloscope screenshot of exemplary measured inverteroutput currents and load voltage during system startup with an RLC load.System dynamics in FIG. 25 show successful synchronization.

Consider load transients in the RLC load where the switch in FIG. 24(a)is opened and closed. FIGS. 26(a) and 26(b) are oscilloscope screenshotsof RLC load transient showing inverter output currents and load voltagewhen the RL load branch is (a) added and (b) removed. The systemdynamics in FIGS. 26(a) and 26(b) correspond to the opening and closingof the switch which interfaces the RL branch of the load, respectively.

FIG. 27 is an oscilloscope screenshot of exemplary measured inverteroutput currents and load voltage in the presence of a diode bridgerectifier load. The system is configured to deliver power to thenonlinear load in FIG. 24(b). Example evidence is presented which showsthat the control is compatible with nonlinear loads. The invertercontrollers are initialized with non-uniform initial conditions as inthe previous two examples. As demonstrated in FIG. 27, the system ofinverters successfully synchronizes and delivers power to the load.

FIGS. 28(a) and 28(b) are oscilloscope screenshots of inverter outputcurrents and load voltage when connected to a diode bridge rectifierload and an inverter is (a) removed and (b) added. In this example, thenumber of parallel inverters connected to the nonlinear load can undergochanges. As illustrated in FIG. 28(a), the remaining inverters maintainsynchronization and deliver power to the load when one inverter isremoved. The pre-synchronization circuit parameters in Table 2 have beenused such that z_(virt)=2κ⁻¹57.5Ω and R_(link)=κ⁻¹100Ω. The inverteraddition transient in FIG. 28(b) is relatively small. Seamless unitaddition can result despite the fact that the actual load is nonlinearand the pre-synchronization virtual load is purely resistive. This canimply that performance during unit addition is not particularlysensitive to the accuracy of the following approximation: choosez_(virt)≈κ⁻¹Nz_(load).

In these examples, the inverter system is delivering power to a pair ofparallel-connected single-phase fans. Each mechanical load is rated for120 V AC operation and the power ratings of each fan were 80 W and 260W. Because an induction machine contains a back electromotive forcevoltage, the fan is not a passive LTI load. Consequently, thesynchronization condition does not apply. However, it can be shown thatthe inverter control still retains the desired performance.

FIG. 29 is an oscilloscope screenshot of exemplary measured inverteroutput currents and load voltage in the presence of a two parallel fanloads. Reusing previously stated initial conditions, the startupperformance of the inverter system can be seen in FIG. 29. Resultsindicate that the inverters successfully synchronize and deliver powerto the mechanical load.

FIGS. 30(a) and 30(b) are oscilloscope screenshots of mechanical loadtransients showing inverter output currents and load voltage during load(a) step-up and (b) load step-down. In this example, the power consumedby the mechanical load is abruptly increased and decreased. FIGS. 30(a)and 30(b) illustrate system dynamics during a step-up and step-down inload power, respectively.

FIG. 31 is a circuit diagram of an exemplary single-phase, constantpower load. A simulation is conducted of a system of inverters which areconfigured to deliver power to a constant power load. An average valuesimulation of the inverter system described above is utilized. Allcontrol parameters and initial conditions are reused for consistency. Asshown in FIG. 31, the load includes a diode bridge rectifier and a 35 Wconstant power load on the dc side. The system of three inverters wasallowed to synchronize and then the load was abruptly activated atapproximately 0.16 s.

FIG. 32 is an oscilloscope screenshot of exemplary simulated invertercurrents, load voltage, and power consumed by the dc load. Asillustrated, the system of inverters successfully delivers power to theload and maintains synchronization. The dc load power is denoted asp_(cpl).

Therefore, a practical method for the implementation of virtualoscillator control is described. After giving the synchronizationcondition for a hardware system of parallel inverters, a designprocedure was described. Furthermore, a method for seamlessly addinginverters into an energized microgrid was described. Example resultswere used to demonstrate the merit of the techniques. Resultsdemonstrate rapid system response to transients and synchronizationdespite non-ideal initial conditions. Seamless addition of invertersinto the energized system can be achieved with the pre-synchronizationmethod. Although the synchronization condition has been proven valid forlinear loads, hardware results indicate that the control performs asdesired with a variety of load types. Control for systems of inverterswith non-identical power ratings can be implemented.

FIG. 33 is a circuit diagram of an exemplary system of inverters withvirtual oscillator control in (a) and exemplary system of coupledoscillators in (b). FIG. 33 illustrates how inverters of heterogeneouspower ratings can be accommodated. the power rating of the j^(th)inverter is denoted by P_(j). The impedance of the j^(th) output filterwill be written as κ_(j) ⁻¹ z_(f)(s), where z_(f)(s) is defined as thereference filter impedance. The former definitions allow us to establisha base impedance, z_(basej)=V_(rated) ²/P_(j), such that theper-unitized j^(th) filter impedance is equal toz_(f)(s)/(κ_(j)z_(basej)). By selecting the per-unit filter impedance ofeach inverter to be identical, it is straightforward to show that

${\frac{P_{j}}{\kappa_{j}} = \frac{P_{k}}{\kappa_{k}}},$∀j,k=1, . . . , N. Hence, current sharing between multiple inverters inproportion to their power ratings is achieved choosing the inverterfilters to have the same per-unit impedance and by incorporating thevalue of κ_(j) into the current scaling as shown in FIG. 33.

FIGS. 34 and 35 are phase diagrams of an exemplary coordinatetransformation of three-phase signals. FIGS. 34 and 35 show (a)three-phase balanced waveforms, (b) space-vector signal in the complexplane, and (c) corresponding waveforms in the αβ-frame. The coordinatetransformation can be used in a controller in several ways. For example,the inverter output currents, i_(abc) (t)=[i_(a)(t), i_(b)(t),i_(c)(t)]^(T), are sensed and transformed to obtain i_(α)(t) andi_(β)(t) with the space vector. Since i_(a)(t)=i_(α)(t) under balancedconditions, the current ιi_(α)(t) is extracted from the virtualoscillator to establish the link with the single-phase equivalent of thethree-phase inverter. To control the switching action of the three-phaseinverter, a set of three-phase modulation signals is generated.

In the quasi harmonic regime, the oscillator terminal voltage can beapproximated as v_(C)(t)=V cos(ωt), where, the amplitude, V, is governedby the choice of σ and R, and the frequency ω≈1/√{square root over(LC)}. Since di_(L)/dt=v_(C), it follows that the current through theinductor in the RLC subcircuit is given by i_(L)(t)=V/(ωL) sin(ωt).Since v_(C)(t) and i_(L)(t) are orthogonal, they can be used to derive aset of three-phase modulation signals. In particular, v_(C) and i_(L)are transformed from the αβ-frame to the abc-frame, multiplied by v, andscaled by the dc-link voltage to yield a set of three-phase modulationsignals, m_(abc)(t)=[m_(a)(t), m_(b)(t), m_(c)(t)]^(T). A pulse widthmodulation scheme can be used to generate the switching signals. Theaverage inverter terminal voltages follow the commanded voltages, e.g.,v_(abc)→v*_(abc). With the approach, the controller state variablescorresponding to the nonlinear oscillator (e.g., v_(C)(t) and i_(L)(t))are directly utilized to generate the three-phase modulation signals.This can eliminate a need for explicit orthogonal-signal generators.

FIG. 36 is a circuit diagram of an exemplary deadzone oscillator forthree-phase signals and FIG. 37 is a circuit diagram of an exemplaryinverter controller for three-phase signals with a pre-synchronizationcircuit. A matrix denoted with the Greek symbol for Xi (Ξ) converts backand forth between a set of three-phase waveforms and a pair of waveformswith a 90 degree phase shift. This allows an interface to the virtualoscillator, which has a naturally existing set of two waveforms withexhibit a 90 degree phase shift, to the external three-phase powersystem. The equations in FIGS. 34 and 35 describe the variabletransformation and the graphs in FIGS. 34 and 35 illustrate what thevariable transformation is doing.

Since the three-phase system can be recast as a single-phase equivalent,the design procedure given above can be applied identically to thethree-phase inverter system.

The following is one implementation that can use the systems and methodsdiscussed above. If a consumer has their own set of resources, theycould utilize a system of power electronics inverters with virtualoscillator control to build a system with minimal design effort. Thetechnology can be especially relevant to renewable energy systems, navaland military installations, and standalone installations in thedeveloping world.

Enabling energy resources of this technology are photovoltaics,batteries, electric vehicles, and fuel-cells. In a microgrid-enabledfuture, consumers could generate and consume their own energy. Sinceenergy is generated and consumed locally, there is a decreaseddependence on the transmission infrastructure and it follows thattransmission losses are decreased.

One potential use of virtual oscillator control involves electricvehicles. In such a setting, an electric vehicle can supply and consumeenergy from a residential microgrid.

Power electronics inverters convert direct current (DC), such as thatproduced by a car battery, into alternating current (AC), the kind ofpower supplied to your home and is used to power home appliances andelectronics. Power electronics inverters utilize switchingsemiconductors and do not require electromechanical energy conversion.

To reduce switching harmonics, filters composed of inductors andcapacitors are utilized such that a low-distortion sinusoidal current isdelivered by an inverter. To improve waveform quality, there is aninherent cost tradeoff between power electronics switching frequency andfilter component size. For instance, higher switching frequencies enablesmaller and cheaper filters but may reduce efficiency.

System and methods are described for controlling a system of multipleinverters which are interconnected in a power system and aredisconnected from the grid utility. The systems and methods of inverterscan provide uninterruptible power to a set of AC loads. The loads can beresidential, commercial, military, industrial, etc. in nature. Thesystem and methods can control the inverters such that the inverters donot require any communication and only require measurements readilyavailable at their AC output terminals. Inverters can automaticallysynchronize upon being added to and removed from the system. The systemsand methods are generalized with respect to the number of inverters, N.

Benefits of the systems and methods include: i) communication is notrequired between inverters, ii) inverters can be added and removed fromthe system during operation, iii) the AC frequency and voltage ismaintained within desired bounds iv) each inverter provides a fractionof power to the load in direct proportion to its power rating. Due tothe absence of a communication network and system-level controller, thesystems and methods do not contain a single point of failure. Theinverters with the control systems and methods provide can reliablepower to the set of loads on the on a network. The systems and methodsprovide a condition for inverter synchronization. If the synchronizationcondition, which depends on the control and inverter parameters, issatisfied, the system of inverters can be guaranteed to synchronizeirrespective of the load type or number of inverters in the system.

The controller can be implemented on any digital platform such as amicrocontroller, digital signal processor, field programmable gatearray, etc. Alternatively, the virtual oscillator can also beconstructed using an analog integrated circuit. The analog circuit canbe configured to control the switching of power electronic semiconductordevices and to process a scaled current and voltage at an output of thepower electronics inverter. The analog circuit can determine anoscillator voltage based on the current. The switches configured to bemanipulated based on an analog oscillator voltage. An analog oscillatorcan be configured to determine the oscillator voltage

The controller processes a set of measurements and generates an ACvoltage command for the inverter hardware. Since no communication isrequired the systems and methods can enhance modularity and inverterscan be seamlessly added to and removed from the network. Repairs can bemade to the system of inverters and the system expanded without a needto reconfigure the system, e.g., the system of inverters canself-organize. The systems and methods are reliable and the controlimplementation is relatively simple to accomplish. The system andmethods enable a bottom-up approach to systems design, e.g., that can beused by military installations, for uninterrupted power on power gridseven during disturbances on the power grid, building installations,medical installations.

The systems, methods, devices, and logic described above may beimplemented in many different ways in many different combinations ofhardware, software or both hardware and software. For example, all orparts of the system may include circuitry in a controller, amicroprocessor, or an application specified integrated circuit (ASIC),or may be implemented with discrete logic or components, or acombination of other types of analog or digital circuitry, combined on asingle integrated circuit or distributed among multiple integratedcircuits. All or part of the logic described above may be implemented asinstructions for execution by a processor, controller, or otherprocessing device and may be stored in a tangible or non-transitorymachine-readable or computer-readable medium such as flash memory,random access memory (RAM) or read only memory (ROM), erasableprogrammable read only memory (EPROM) or other machine-readable mediumsuch as a compact disc read only memory (CDROM), or magnetic or opticaldisk. Thus, a product, such as a computer program product, may include astorage medium and computer readable instructions stored on the medium,which when executed in an endpoint, computer system, or other device,cause the device to perform operations according to any of thedescription above.

The processing capability of the system may be distributed amongmultiple system components, such as among multiple processors andmemories, optionally including multiple distributed processing systems.Parameters, databases, and other data structures may be separatelystored and managed, may be incorporated into a single memory ordatabase, may be logically and physically organized in many differentways, and may implemented in many ways, including data structures suchas linked lists, hash tables, or implicit storage mechanisms. Programsmay be parts (e.g., subroutines) of a single program, separate programs,distributed across several memories and processors, or implemented inmany different ways, such as in a library, such as a shared library(e.g., a dynamic link library (DLL)). The DLL, for example, may storecode that performs any of the system processing described above.

Many modifications and other embodiments set forth herein can come tomind based on the teachings presented in the foregoing descriptions andthe associated drawings. Although specified terms are employed herein,they are used in a generic and descriptive sense only and not forpurposes of limitation.

The invention claimed is:
 1. A system, comprising: an oscillatorcontroller, the oscillator controller including a resistor, a capacitor,an inductor, and a piecewise linear voltage-dependent current sourceconnected in parallel; and power electronics inverters connected withthe oscillator controller in a network, the oscillator controller togenerate an oscillating signal to modulate the power electronicsinverters; wherein the power electronics inverters utilize measurementsat local terminals, without a need to exchange information between otherpower electronics inverters for synchronization.
 2. The system of claim1, wherein the power electronics inverters oscillate as a synchronizedalternating current system.
 3. The system of claim 1, wherein voltagelimits are respected across a no-load to a maximum rated load range. 4.The system of claim 1, wherein a distortion of a sinusoidal output ofthe power electronics inverters is reduced.
 5. The system of claim 1,wherein the power electronics inverters automatically synchronize acoutputs across the network.
 6. The system of claim 5, wherein thesynchronization occurs automatically upon adding or removing a powerelectronics inverter to the network.
 7. The system of claim 5, whereinthe synchronization occurs without a phase lock loop to generatesinusoidal reference waveforms or explicit communication between powerelectronics inverters.
 8. The system of claim 5, wherein thesynchronization occurs without a proportional-integral orproportional-integral-derivate controller.
 9. The system of claim 5,wherein the synchronization is agnostic to a number of power electronicsinverters and loads.
 10. The system of claim 1, wherein the powerelectronics inverters further include the oscillator controllerimplemented as code stored in memory and executed by a processor or asan analog integrated circuit.
 11. The system of claim 1, wherein thepower electronics inverters share a load proportional to their size. 12.The system of claim 1, wherein real and reactive power need not becalculated in real time.
 13. The system of claim 1, wherein the powerelectronics inverters further include a pre-synchronization circuit tosynchronize the other power electronics inverters before adding them toan energized network.
 14. A system, comprising: a power electronicsinverter including switching semiconductor devices and passive filteringcomponents; at least one of a microcontroller to control the switchingsemiconductor devices, the microcontroller configured to digitize thecurrent and voltage at an output of the power electronics inverter anddetermine an oscillator voltage based on the current, the switchingsemiconductor devices being manipulated based on the oscillator voltage,or an analog circuit configured to control the switching semiconductordevices, the analog circuit configured to process a scaled current andvoltage at the output of the power electronics inverter and determine anoscillator voltage based on the current, the switching semiconductordevices configured to be manipulated based on the analog oscillatorvoltage; and an oscillator controller of the microcontroller, theoscillator controller including a resistor, a capacitor, an inductor,and a piecewise linear voltage-dependent current source connected inparallel, wherein the oscillator controller is configured to determinethe oscillator voltage.
 15. The system of claim 14, wherein theswitching semiconductor devices are interleaved to reduce operatingripple.
 16. The system of claim 14, wherein the oscillator controllercomprises an analog oscillator to determine the oscillator voltage. 17.The system of claim 14, wherein the microprocessor or analog circuitfurther includes a pulse width modulator to send a signal to the powerelectronics switching semiconductor devices to control the switchingbased on the oscillator voltage.
 18. A method, comprising: an oscillatorcontroller including a resistor, a capacitor, an inductor, and apiecewise linear voltage-dependent current source connected in parallel,an analog to digital converter and accompanying analog circuitry fordetermining an output current of a power electronics inverter, for:determining an instantaneous oscillation voltage based on the current;and controlling switching of the power electronics circuit based on theinstantaneous oscillation voltage.
 19. The method of claim 18, whereinthe analog circuitry comprises an integrated circuit for determining theoutput current of the power electronics inverter.
 20. The method ofclaim 18, further comprising inputting a scaled current into theoscillator controller to determine the oscillation voltage.